The first thing I'd check is that it may be either GITHUB_TOKEN or GH_TOKEN.

gh help environment says GH_TOKEN and GITHUB_TOKEN are both recognized, with GH_TOKEN taking precedence.

So if gh auth login says:

The value of the GITHUB_TOKEN environment variable is being used for authentication.

then one of the likely issues is:

• you are clearing the variable in the wrong shell syntax
• or you actually have GH_TOKEN / GITHUB_TOKEN set in a parent process / wrapper shell

For example:

• in PowerShell, clear it with:

Remove-Item Env:GITHUB_TOKEN -ErrorAction Ignore
Remove-Item Env:GH_TOKEN -ErrorAction Ignore

or:

$env:GITHUB_TOKEN = $null
$env:GH_TOKEN = $null

• in cmd.exe, use:

set GITHUB_TOKEN=
set GH_TOKEN=

If you're using Git Bash / another wrapper, that shell may have inherited the variable from elsewhere even if your env check is misleading.

So my short answer is: check both GH_TOKEN and GITHUB_TOKEN, and clear them using the syntax for the shell you're actually running gh in.